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Math Help - Irreducible Polynomials:

  1. #1
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    Irreducible Polynomials:

    (1):
    Find all irreducible polynomials of the form x^2 + ax +b , where a,b belong to the field \mathbb{F}_3 with 3 elements.
    Show explicitly that \mathbb{F}_3(x)/(x^2 + x + 2) is a field by computing its multiplicative monoid.
    Identify [ \mathbb{F}_3(x)/(x^2 + x + 2)]* as an abstract group.

    any suggestions please?
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  2. #2
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    Quote Originally Posted by joanne_q View Post
    (1):
    Find all irreducible polynomials of the form x^2 + ax +b , where a,b belong to the field \mathbb{F}_3 with 3 elements.
    Just list all of them x^2 + x+1,x^2+x+2,... there are only 9 or them. Now, just check which ones have zeros (there are only three zeros to check). And this tells you which are irreducible and which are not.

    Show explicitly that \mathbb{F}_3(x)/(x^2 + x + 2) is a field by computing its multiplicative monoid.
    Identify [ \mathbb{F}_3(x)/(x^2 + x + 2)]* as an abstract group.
    Any element in \mathbb{F}_3[x]/(x^2+x+2) has form a+bx+(x^2+x+2). Now show that these elements form a field.
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